have that ratio to a greater sphere LMN: therefore, by inversion, the sphere LMN has to the sphere ABC the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, so is the sphere DEF to some sphere, which must be less (14. 5.) than the sphere ABC, because the sphere LMN is greater than the sphere DEF. Therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC; which was shown to be impossible therefore the sphere ABC has not to any sphere greater than DEF the triplicate ratio of that which BC has to EF: and it was demonstrated, that neither has it that ratio to any sphere less than DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D.

FINIS.

NOTES,

CRITICAL AND GEOMETRICAL;

CONTAINING

AN ACCOUNT OF THOSE THINGS IN WHICH THIS EDITION DIFFERS FROM THE GREEK TEXT;

AND

THE REASONS OF THE ALTERATIONS WHICH HAVE BEEN MADE.

AS ALSO

OBSERVATIONS

ON SOME OF THE PROPOSITIONS.

BY ROBERT SIMSON, M. D.

EMERITUS PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW.

NOTES,

CRITICAL AND GEOMETRICAL;

CONTAINING

AN ACCOUNT OF THOSE THINGS IN WHICH THIS EDITION DIFFERS FROM THE GREEK TEXT;

AND

THE REASONS OF THE ALTERATIONS WHICH HAVE BEEN MADE.

AS ALSO

OBSERVATIONS

on some of the PROPOSITIONS.

BY ROBERT SIMSON, M. D.

EMERITUS PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF GLASGOW.